In the xy-plane, a line has a y-intercept of -{6} and an x-intercept of 3. Which of the following is...

1. TRANSLATE the intercept information into coordinates

• Given information:
  • y-intercept = -6
  • x-intercept = 3

• What this tells us:
  • y-intercept means the line crosses the y-axis at (0, -6)
  • x-intercept means the line crosses the x-axis at (3, 0)

2. INFER the best approach

• With two points, we can find the slope and use slope-intercept form
• Strategy: Calculate slope, then write y = mx + b

3. SIMPLIFY to find the slope

Using the slope formula with points (0, -6) and (3, 0):

\(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)

\(\mathrm{m = \frac{0 - (-6)}{3 - 0}}\)

\(\mathrm{m = \frac{6}{3}}\)

\(\mathrm{m = 2}\)

4. INFER the complete equation

• We have: slope \(\mathrm{m = 2}\) and y-intercept \(\mathrm{b = -6}\)
• Using slope-intercept form: \(\mathrm{y = 2x - 6}\)

5. APPLY CONSTRAINTS to verify our answer

• Check at (0, -6): \(\mathrm{y = 2(0) - 6 = -6}\) ✓
• Check at (3, 0): \(\mathrm{y = 2(3) - 6 = 0}\) ✓

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the meaning of intercepts and create incorrect coordinate points.

Some students think 'y-intercept of -6' means the point (-6, 0) instead of (0, -6), or 'x-intercept of 3' means (0, 3) instead of (3, 0). When they use the wrong points, their slope calculation gives them \(\mathrm{m = -2}\) instead of \(\mathrm{m = 2}\).

This leads them to select Choice A (\(\mathrm{y = -2x - 6}\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating the slope.

They correctly identify the points (0, -6) and (3, 0), but when computing the slope, they calculate: \(\mathrm{m = \frac{0-(-6)}{3-0} = \frac{-6}{3} = -2}\), forgetting that subtracting a negative becomes positive. Or they might compute (0+6)/(3-0) but then incorrectly apply the sign.

This also leads them to select Choice A (\(\mathrm{y = -2x - 6}\)).

The Bottom Line:

This problem tests whether students truly understand what intercepts represent as coordinate points, and whether they can accurately perform the slope calculation with negative values. The key insight is that intercepts always have one coordinate equal to zero.